Asymptotic expansion of the minimum covariance determinant estimators

In Cator and Lopuhaä (arXiv:math.ST/0907.0079) [3], an asymptotic expansion for the minimum covariance determinant (MCD) estimators is established in a very general framework. This expansion requires the existence and non-singularity of the derivative in a first-order Taylor expansion. In this paper, we prove the existence of this derivative for general multivariate distributions that have a density and provide an explicit expression, which can be used in practice to estimate limiting variances. Moreover, under suitable symmetry conditions on the density, we show that this derivative is non-singular. These symmetry conditions include the elliptically contoured multivariate location-scatter model, in which case we show that the MCD estimators of multivariate location and covariance are asymptotically equivalent to a sum of independent identically distributed vector and matrix valued random elements, respectively. This provides a proof of asymptotic normality and a precise description of the limiting covariance structure for the MCD estimators.